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Baseball and Physics

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  1. 1927 Yankees: Greatest baseball team ever assembled 1927 Solvay Conference: Greatest physicsteam ever assembled Baseball and Physics MVP’s

  2. #521, September 28, 1960 Hitting the Baseball “...the most difficult thing to do in sports” --Ted Williams BA: .344 SA: .634 OBP: .483‡ HR: 521 ‡career record

  3. Introduction: Description of Ball-Bat Collision • forces large (>8000 lbs!) • time is short (<1/1000 sec!) • ball compresses, stops, expands • kinetic energy  potential energy • bat compresses ball….ball bends bat • hands don’t matter! • GOAL: maximize ball exit speed vf • vf 105 mph  x  400 ft x/vf = 4-5 ft/mph How to predict vf?

  4. Bat Rest Frame “Lab” Frame vball vbat vrel vf eAvrel Kinematics: Reference Frames vf =eAvball +(1+eA)vbat • eA  “Apparent Coefficient of Restitution” = “BESR” - 0.5 • property of ball & bat • weakly dependent on vrel •  0.2  vf 0.2 vball + 1.2 vbat Conclusion:vbatmuch more important than vball

  5. r bat recoil factor = mball/mbat,eff e “Coefficient of Restitution” Kinematics: Conservation Laws (Accounting for eA) v m1 m2 eAv m1

  6. . . . CM b Kinematics: bat recoil factor • typical numbers • mball = 5.1 oz • mbat = 31.5 oz • k = 9.0 in • b = 6.3 in • r = .24 • e = 0.5 • eA = 0.21 = +  0.16 x 1.49  0.24 • All things equal, want r small • But….

  7. All things are not equal • Mass & Mass Distribution affect bat speed • Conclusion: • mass of bat matters….but probably not a lot

  8. “bounciness” of ball Kinematics: Coefficient of Restitution (e): (Energy Dissipation) • in CM frame: Ef/Ei = e2 • massive rigid surface: e2 = hf/hi • typically e  0.5 • ~3/4 CM energy dissipated! • probably depends on impact speed • depends on ball and bat!

  9. COR: Is the Ball “Juiced”? • MLB:e = 0.546  0.032 @ 58 mph on massive rigid surface For ball on stationary bat:

  10. CM 12 10 8 6 4 2 0 -2 0 5 10 15 20 25 30 Putting it all together….. vf = eA vball + (1+eA) vbat

  11. CM More Realistic Analysis vf = eA vball + (1+eA) vbat

  12. III. Dynamics Model for Ball-Bat Colllision: Accounting for Energy Dissipation • Collision excites bending vibrations in bat • Ouch!! Thud!! • Sometimes broken bat • Energy lost  lower vf (lower e) • Bat not rigid on time scale of collision • What are the relevant degrees of freedom? see AMN, Am. J. Phys, 68, 979 (2000)

  13. The Essential Physics: A Toy Model bat ball Mass= 1 2 4 rigid  << 1 m on Ma (1 on 2)  >> 1 m on Ma+Mb (1 on 6) flexible

  14. A Dynamic Model of the Bat-Ball Collision y 20 Euler-Bernoulli Beam Theory‡ y z • Solve eigenvalue problem for free oscillations (F=0) •  normal modes(yn, n) • Model ball-bat force F • Expand y in normal modes • Solve coupled equations of motion for ball, bat ‡Note for experts: full Timoshenko (nonuniform) beam theory used

  15. f1 = 177 Hz f3 = 1179 Hz f2 = 583 Hz f4 = 1821 Hz nodes Normal Modesof the Bat Louisville Slugger R161 (33”, 31 oz) Can easily be measured (modal analysis)

  16. Measurements via Modal Analysis Louisville Slugger R161 (33”, 31 oz) FFT frequencybarrel node ExptCalcExptCalc 17917726.526.6 58258327.828.2 1181117929.029.2 1830182130.029.9 Conclusion: free vibrations of bat can be well characterized

  17. F=kxn F=kxm Model for the Ball 3-parameter problem: k nv-dependence of  m COR of ball with rigid surface

  18. Putting it all together…. ball compression • Procedure: • specify initial conditions • numerically integrate coupled equations • find vf = ball speed after ball and bat separate

  19. General Result energy in nth mode Fourier transform Conclusion:only modes with fn  < 1 strongly excited

  20. Results: Ball Exit Speed Louisville Slugger R161 33-inch/31-oz. wood bat only lowest mode excited lowest 4 modes excited Conclusion:essential physics under control

  21. CM nodes Application to realistic conditions: (90 mph ball; 70 mph bat at 28”)

  22. The “sweet spot” 1. Maximum vf (~28”) 2. Minimum vibrational energy (~28”) 3. Node of fundamental (~27”) 4. Center of Percussion (~27”) 5. “don’t feel a thing”

  23. 3 Displacement at 5” 2 1 y (mm) 0 -1 impact at 27" -2 -3 0 0.5 1 1.5 2 t (ms) Boundary conditions • Conclusions: • size, shape, boundary conditions at far end don’t matter • hands don’ t matter!

  24. T= 0-1 ms Time evolution of the bat T= 1-10 ms

  25. Wood versus Aluminum • Kinematics • Length, weight, MOI “decoupled” • shell thickness, added weight • fatter barrel, thinner handle • Weight distribution more uniform • ICM larger (less rot. recoil) • Ihandle smaller (easier to swing) • less mass at contact point • Dynamics • Stiffer for bending • Less energy lost due to vibrations • More compressible • COReff larger

  26. tennis ball/racket Effect of Bat on COR: Local Compression • CM energyshared between ball and bat • Ball inefficient:  75% dissipated • Wood Bat • kball/kbat ~ 0.02 • 80% restored • eeff = 0.50-0.51 • Aluminum Bat • kball/kbat ~ 0.10 • 80% restored • eeff = 0.55-0.58 Ebat/Eball kball/kbat  xbat/ xball >10% larger!

  27. Wood versus Aluminum: Dynamics of “Trampoline” Effect “bell” modes: “ping” of bat • Want k small to maximize stored energy • Want >>1 to minimize retained energy • Conclusion: there is an optimum 

  28. Performance Comparison

  29. Things I would like to understand better • Relationship between bat speed and bat weight and weight distribution • Location of “physiological” sweet spot • Better model for the ball • Better understanding of trampoline effect for aluminum bat • Why is softball bat different from baseball bat? • Effect of “corking” the bat

  30. Summary & Conclusions • The essential physics of ball-bat collision understood • bat can be well characterized • ball is less well understood • the “hands don’t matter” approximation is good • Vibrations play important role • Size, shape of bat far from impact point does not matter • Sweet spot has many definitions • Aluminum outperforms wood!